Advanced downhole waveform interpretation

ABSTRACT

Systems and methods process a measured ultrasonic response waveform to determine a well casing thickness and an acoustic impedance of a sealing medium surrounding the well casing. An array of simulated response waveforms corresponding to a set of candidate acoustic impedances for the sealing medium surrounding the well casing and a set of candidate well casing thicknesses is generated. A simulated response waveform from the array of simulated response waveforms is identified that best matches the measured response waveform so as to determine the sealing medium acoustic impedance.

CROSS-REFERENCES TO RELATED APPLICATIONS

The present application claims the benefit under 35 USC § 119(e) of U.S.Provisional Appln. No. 62/088,958 filed Dec. 8, 2014; the fulldisclosure which is incorporated herein by reference in its entirety forall purposes.

BACKGROUND

The process for creating a well (e.g., oil, natural gas) typicallyincludes drilling a hole through a ground formation, inserting a wellcasing into the hole, and filling the resulting annulus between theinterior surface of the drilled hole and the exterior surface of thewell casing with a suitable sealing medium (e.g., cement). The sealingmedium serves to prevent flow of a liquid (e.g., oil) and/or a gas(e.g., natural gas) along the annulus so as to inhibit and preferablyprevent leakage of the liquid and/or gas via the annulus. Typically, thesealing medium is injected into the annulus at an end of a length of thewell casing and flows longitudinally along the annulus to fill theannulus along the length of the well casing.

Unfortunately, the injection of the sealing medium may sometimes resultin an annulus that is inadequately sealed. Because an inadequatelysealed annulus can result in significant detrimental leakage along theannulus, it is important that any inadequately sealed portions of theannulus be identified and fixed. Ultrasonic inspection is typically usedto evaluate if the annulus is adequately sealed. An ultrasonicinspection tool is moved along the inside of the well casing, transmitsan ultrasonic pulse towards the inside wall of the well casing, andgenerates an output signal in response to the returning ultrasonicwaves. Analysis of the output signal is performed to evaluate, in part,the acoustic impedance of the sealing medium in the annulus. Theacoustic impedance of the sealing medium can then be assessed toevaluate if the annulus at the inspected location is adequately orinadequately sealed. The process is repeated along the depth of the wellcasing and repeated at different azimuth angles to sufficiently inspectthe annulus.

BRIEF SUMMARY

The following presents a simplified summary of some embodiments of theinvention in order to provide a basic understanding of the invention.This summary is not an extensive overview of the invention. It is notintended to identify key/critical elements of the invention or todelineate the scope of the invention. Its sole purpose is to presentsome embodiments of the invention in a simplified form as a prelude tothe more detailed description that is presented later.

Improved approaches and related systems are provided for processing ameasured response waveform generated by a downhole ultrasonic inspectiontool used to inspect a well. Simulated response waveforms are generatedbased on the measured response waveform. Each of the simulated responsewaveforms can be further based on any suitable combination of fourindependent parameters: a respective candidate well casing thickness, arespective candidate acoustic impedance for a sealing medium in theannulus around the well casing, a respective candidate annulusthickness, and a respective candidate acoustic impedance for the groundformation surrounding the annulus (i.e., inner surface of the hole). Thesimulated response waveforms are compared to the measured responsewaveform to identify which of the simulated response waveforms bestmatches the measured response waveform, thereby identifying a best-fitwell casing thickness, a best-fit acoustic impedance for the annulussealing medium, a best fit thickness for the annulus thickness, and/or abest fit acoustic impedance for the ground formation surrounding theannulus. In many cases, the sealing medium in the annulus can beconsidered infinite in extent so that the matching procedure only usestwo free parameters (casing thickness and acoustic impedance of thesealing medium in the annulus). The casing thickness and acousticimpedance of the sealing medium in the annulus change the acousticsignal in very different ways and, therefore, the fitting procedureproduces the optimal estimate of the acoustic impedance of the annulussealing medium independent of the casing thickness. For other cases,especially when the annulus is thin, computing the best fit using allfour parameters produces estimates of the casing thickness and theacoustic impedance of the sealing medium in the annulus independent ofthe annulus thickness and the acoustic impedance of the ground formationsurrounding the annulus. The signal processing described herein cangreatly reduce the influence of the third interface (between the annulussealing medium and the ground formation surrounding the annulus) onestimates of the acoustic impedance of the sealing medium in theannulus, an advantage not provided by prior approaches. The resultingestimates of well casing thickness and surrounding annulus sealingmedium acoustic impedance have been found to be significantly moreaccurate that those generated by prior approaches, thereby increasingconfidence in the inspection results and providing better informationfor decisions as to whether the annulus is adequately or inadequatelysealed.

Thus, in one aspect, a computer-implemented method of processing ameasured ultrasonic response waveform to determine a thickness of a wellcasing and an acoustic impedance of a sealing medium in an annulussurrounding the well casing is provided. The method includes processinga measured response waveform resulting from an ultrasonic signaltransmitted toward the well casing from within the well casing todetermine an arrival time for a first reflection from the well casing.An array of simulated response waveforms is generated corresponding tothe arrival time, a set of candidate acoustic impedances for the sealingmedium in the annulus, a set of candidate well casing thicknesses, a setof candidate annulus thicknesses, and a set of candidate acousticimpedances for a ground formation surrounding the annulus. A best-fitsimulated response waveform that best matches the measured responsewaveform is identified from the array of simulated response waveforms soas to identify a best-fit well casing thickness of the set of candidatewell casing thicknesses, a best-fit sealing medium acoustic impedance ofthe set of candidate acoustic impedances for the sealing medium in theannulus, a best-fit annulus thickness of the set of candidate annulusthicknesses, and a best-fit acoustic impedance for the ground formationsurrounding the annulus of the set of candidate acoustic impedances forthe ground formation surrounding the annulus associated with thebest-fit simulated response waveform. The method can include calculatingthe best-fit well casing thickness, the best-fit sealing medium acousticimpedance, the best-fit annulus thickness, and the best-fit acousticimpedance for the ground formation surrounding the annulus for aplurality of depths and a plurality of azimuth angles.

In many embodiments, a portion of the measured response waveform isprocessed to determine one or more parameters used to generate the arrayof simulated response waveforms. For example, the method can includeprocessing a portion of the measured response waveform to determine anapproximate thickness of the well casing and selecting the set ofcandidate well casing thicknesses based on the approximate thickness ofthe well casing. As another example, the method can include processing aportion of the measured response waveform to determine a transducerimpulse response for an ultrasonic transducer used to generate themeasured response waveform. In many embodiments, each of the array ofsimulated response waveforms is based on the transducer impulseresponse.

Each of the simulated response waveforms can be generated using anysuitable approach. For example, generating each of the array ofsimulated response waveforms can include: (a) calculating amplitudes forsimulated echo waves returning to the ultrasonic transducer from thewell casing; (b) calculating amplitudes for simulated echo wavesreturning to the ultrasonic transducer from an interface between thesealing medium in the annulus and the ground formation surrounding theannulus; (c) generating corrected amplitudes for the simulated echowaves by correcting the amplitudes and pulse shape for the simulatedecho waves to account for diffraction, refraction, cylindrical wallgeometry, and resulting beam size at a transducer plane of theultrasonic transducer; (d) generating a first-iteration simulatedwaveform based on the corrected amplitudes; and (e) convolving thefirst-iteration simulated waveform with the transducer impulse responseto generate the respective simulated response waveform. Each of thearray of simulated response waveforms can be based on reflection andtransmission coefficients for an inner surface of the well casing, anouter surface of the well casing, the outer surface of the well (i.e.outer surface of the medium in the annulus), and different orderreverberations in the well casing.

In many cases encountered in ultrasonic inspection of wells, the annulusthickness is sufficiently large such that reflections of the ultrasonicsignal from the annulus-well boundary interface (e.g., sealingmedium/ground formation interface) do not significantly impact theresponse waveform measured by the ultrasonic transducer. For these casesthe annulus thickness can be considered infinite and the matchingprocedure can use only two free parameters: well casing thickness andthe acoustic impedance for the sealing medium in the annulus around thewell casing. For some cases where the annulus is thin enough such thatreflections of the ultrasonic signal from the sealing medium/groundformation boundary do significantly impact the measured responsewaveform, all four parameters can be used to produce candidate waveformsthat better match the measured response waveform. By accounting forreflections of the ultrasonic signal from the sealing medium/groundformation boundary, more accurate estimates of well casing thickness andacoustic impedance for a sealing medium in the annulus can be attained.In the description that follows, details of a matching procedure thatemploys a two-parameter model are presented. Details of a matchingprocedure that employs a four-parameter model are also presented bydescribing changes for the four-parameter model relative to thetwo-parameter model. The approaches described herein provide robustestimates of the well casing thickness and the acoustic impedance forthe sealing medium in the annulus.

The computer-implemented method of processing the measured ultrasonicresponse waveform can include a second-iteration that may improve howclosely a best-fit simulated response waveform matches the measuredresponse waveform, thereby improving accuracy of the well casingthickness estimate and the sealing medium acoustic impedance estimate.For example, the method can include: (a) calculating a differentialsignal between the measured response waveform and the best-fit simulatedresponse waveform; (b) calculating a corrected transducer impulseresponse for the ultrasonic transducer by correcting the transducerimpulse response based on the differential signal; (c) generating asecond-iteration array of simulated response waveforms based on thecorrected transducer impulse response and corresponding to the signalarrival time, a second-iteration set of candidate acoustic impedancesfor the annulus sealing medium surrounding the well casing, asecond-iteration set of candidate well casing thicknesses, asecond-iteration set of candidate annulus thicknesses, and asecond-iteration set of candidate acoustic impedances for the groundformation surrounding the annulus; and (d) identifying asecond-iteration best-fit simulated response waveform from thesecond-iteration array of simulated response waveforms that best matchesthe measured response waveform so as to identify a second-iterationbest-fit well casing thickness of the second-iteration set of candidatewell casing thicknesses, a second-iteration best-fit surrounding sealingmedium acoustic impedance of the second-iteration set of candidateacoustic impedances for the sealing medium in the annulus, asecond-iteration best-fit annulus thickness of the second-iteration setof candidate annulus thicknesses, and a second-iteration best-fitacoustic impedance for the ground formation surrounding the annulus ofthe second-iteration set of candidate acoustic impedances for the groundformation surrounding the annulus associated with the second-iterationbest-fit simulated response waveform. Generating each of thesecond-iteration array of simulated response waveforms can includeconvolving a simulated response waveform with the corrected transducerimpulse response to generate the respective simulated response waveformof the second-iteration array of simulated response waveforms.

The computer-implemented method of processing the measured ultrasonicresponse waveform can include generating a confidence value based ondeviation between the best-fit simulated response waveform and themeasured response waveform. The confidence value can be evaluated aloneor in combination with the best-fit well casing thickness and thebest-fit sealing medium acoustic impedance. For example, the method caninclude processing the best-fit well casing thickness, the best-fitsealing medium acoustic impedance, and the confidence value incombination to produce a final confidence value indicative of whetherthe actual sealing medium acoustic impedance is within a designatedrange. The method can include calculating the best-fit well casingthickness, the best-fit sealing medium acoustic impedance, and the finalconfidence value for a plurality of depths and a plurality of azimuthangles.

In another aspect, a system is provided for processing a measuredultrasonic response waveform to determine a well casing thickness and anacoustic impedance of a sealing medium in the annulus surrounding thewell casing. The system includes a processor and a tangible memorystoring non-transient instructions executable by the processor to causethe processor to perform the computer-implemented method of processingthe measured ultrasonic response waveform to determine a well casingthickness and an acoustic impedance of the sealing medium in the annulussurrounding the well casing described herein.

For a fuller understanding of the nature and advantages of the presentinvention, reference should be made to the ensuing detailed descriptionand accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified schematic drawing illustrating inspection of awell using a down-hole ultrasonic inspection tool, in accordance withmany embodiments.

FIG. 2 shows an example response waveform measured by a down-holeultrasonic inspection tool.

FIG. 3 is a simplified schematic diagram of a method of processing ameasured ultrasonic response waveform to determine a well casingthickness and an acoustic impedance of an annulus sealing mediumsurrounding the well casing, in accordance with many embodiments.

FIG. 4 shows a frequency spectrum of a measured response waveform thatcan be used to identify a characteristic frequency used to estimate wellcasing thickness, in accordance with many embodiments.

FIG. 5 illustrates a portion of the example measured response waveformof FIG. 1 that can be used to approximate a transducer impulse response,in accordance with many embodiments.

FIG. 6 shows an example response waveform to a delta-function ultrasonicprobe signal that can be modified to account for diffraction,refraction, and the resulting size of the ultrasound beam at thetransducer plane to generate a simulated response waveform correspondingto a candidate well casing thickness and a candidate acoustic impedancefor the sealing medium in the annulus surrounding the well casing, inaccordance with many embodiments.

FIG. 7 shows an example first-iteration simulated response waveformgenerated by convolving the response waveform of FIG. 6 with thetransducer impulse response of FIG. 5, in accordance with manyembodiments.

FIG. 8 shows an overlay comparing the measured response waveform of FIG.2 with the example first-iteration simulated response waveform of FIG.7, in accordance with many embodiments.

FIG. 9 is a simplified schematic diagram of acts for a second iterationin the method of FIG. 3, in accordance with many embodiments.

FIG. 10A shows an overlay comparing the measured response waveform ofFIG. 2 with the example first-iteration simulated response waveform ofFIG. 7 and a second-iteration simulated response waveform, in accordancewith many embodiments.

FIG. 10B and FIG. 10C show overlays comparing simulated responsewaveforms for an infinitely thick cement-filled annulus surrounding awell casing and cement-filled annuluses having two different finitethicknesses, in accordance with many embodiments.

FIG. 11 shows a resulting contour plot of surrounding annulus sealingmedium acoustic impedance for a range of depths and azimuth angles basedon plane wave approximation based simulated response waveforms, inaccordance with many embodiments.

FIG. 12 illustrates how two-dimensional effects associated with thecylindrical well casing can change the direction of the waves such thathigher-order reflections miss the transducer.

FIG. 13 is a simplified schematic diagram of acts for generating asimulated response waveform that accounts for two-dimensional effects,in accordance with many embodiments.

FIG. 14 is a simplified schematic diagram of a method of processing ameasured ultrasonic response waveform to determine a well casingthickness and an acoustic impedance of a sealing medium in the annulussurrounding the well casing, in accordance with many embodiments.

FIG. 15 shows a resulting contour plot of acoustic impedance of theannulus sealing medium surrounding the casing for a range of depths andazimuth angles based on simulated response waveforms that account fortwo-dimensional effects, in accordance with many embodiments.

DETAILED DESCRIPTION

In the following description, various embodiments of the presentinvention will be described. For purposes of explanation, specificconfigurations and details are set forth in order to provide a thoroughunderstanding of the embodiments. It will also, however, be apparent toone skilled in the art that the present invention may be practicedwithout the specific details. Furthermore, well-known features may beomitted or simplified in order not to obscure the embodiment beingdescribed.

Referring now to the drawings, in which like reference numeralsrepresent like parts throughout the several views, FIG. 1 illustratesinspection of a well 12 using a down-hole ultrasonic inspection tool 14,in accordance with many embodiments. The well 12 is formed by drilling ahole through a ground formation 16 and includes a well casing 18inserted into the hole thereby forming a surrounding annulus between anouter surface 20 of the well casing 18 an inner surface 22 of the hole.A sealing medium 24 (e.g., cement) is injected into the annulus andcured to block flow of liquid and/or gas along the annulus, therebyinhibiting and preferably preventing well leakage via the annulus.

The down-hole ultrasonic inspection tool 14 includes an ultrasonictransducer 26 positioned inside the casing at a certain position fromits center. The transducer sends a probe ultrasound signal (aa) towardsthe casing wall. The probe ultrasound signal (aa) is partially reflectedback from the inner casing interface 32 and is partially propagatedinside the casing. Partial reflections (bb, cc, dd) and transmissions(ee and ff) occur at all the interfaces of the layered structure of thewell (i.e., mud/casing 18, casing 18/sealing medium 24, sealing medium24/ground formation 16) and result in a long chain of back-reflectedultrasound transients (including multiple reverberations within thestructural layers), contributing to the measured response waveform 50.The measured response waveform 50 is detected by the transducer 26 andcan be analyzed with a numerical procedure to determine an acousticimpedance for the sealing medium 24 in the annulus surrounding the wellcasing 18, a well casing thicknesses, a thicknesses of the annulus, andan acoustic impedance for the ground formation surrounding the annulus.

FIG. 2 shows an example measured response waveform 50 generated by thedown-hole ultrasonic inspection tool 14 for the case where the annulusis thick enough to ignore the reflection from the sealing medium24/ground formation 16 interface. The detected response waveform is theresult of reverberation of the initial (probe) ultrasonic signal insidethe casing 18 with partial transmission of waves back to the transducer26. In many embodiments, a simulated signal is analytically formed tofit the measured response waveform 50. The measured response waveform 50includes a first large transient (“zero-order” reflection 52)corresponding to the initial reflection wave, a second transient(“first-order” reflection 54), as well as subsequent higher ordertransients corresponding to respective subsequent higher orderreflections from the well casing outer surface 20.

The progressive decay in the amplitude in the transients is indicativeof the acoustic impedance of the surrounding sealing medium 24. Using aone-dimensional plane wave approximation, a simulated signal can becomputed and compared with the measure response waveform 50 as follows.First, the amplitude of the zero-order reflection 52 is given by:P ₀=1*R ₀₁,  (equation 1)

where R₀₁ is the ultrasonic wave reflection coefficient at a mud/wellcasing 18 interface to account for the presence of mud within the wellcasing 18.

The first-order reflection reaches the transducer 26 after sequentialreflection from the well casing outer surface 20 and transmissionthrough the well casing inner surface 32. Using a one-dimensional planewave approximation, the amplitude of the first-order reflection 54 canbe given by:P ₁=1*R ₁₂(1−R ₁₀)²,  (equation 2)

where R₁₀=−R₀₁ and R₁₀=(z₁−z₀)(z₁+z₀), R₁₂=(z₂−z₁) (z₂+z₁); z₀, z₁ andz₂ are acoustic impedances of mud, the well casing 18 (e.g., steel), andthe surrounding sealing medium 24 (e.g., cement) respectively.Similarly, reflection coefficients can be defined for the boundarybetween the sealing medium 24 in the annulus and the ground formation16.

For high order reflections, using a one-dimensional plane waveapproximation, the amplitudes can be given by:P _(i) =P _(i-1) *R ₁₀ *R ₁₂, where i>2.  (equation 3)

When the annulus is thin enough that the signal reflected from thesealing medium 24/ground formation 16 interface arrives at thetransducer 26 within the detected time window, similar transmission andreflection coefficients can be used to account for the interface betweenthe sealing medium 24 in the annulus and the ground formation 16.

Using equations (1), (2), and (3), and similar equations for the sealingmedium 24/ground formation 16 interface (for the case of a thinannulus), the amplitudes of transients in simulated response waveformscan be generated for a suitable set of candidate acoustic impedances ofthe surrounding sealing medium 24 and the acoustic impedance of theground formation 16. The simulated response waveforms can then becompared to the measured response waveform 50 to identify which one bestmatches the measured response waveform 50 so as to identify which of thecandidate surrounding sealing medium 24 acoustic impedances correspondsto the measured response waveform 50. The interface between thesurrounding sealing medium 24 and the ground formation 16 can also beaccounted for in the determination of the simulated response waveforms.Reflections from the interface between the sealing medium 24 and theground formation 16 will, however, reach the transducer 26 with muchsmaller amplitudes than reflections induced by ultrasonic reverberationswithin the well casing 18. For the case where the annulus is thin enoughto affect the measured response waveform 50, reflection from theinterface between the sealing medium 24 and the ground formation 16 canbe taken into account when generating a simulated response waveform tomatch the measured response waveform 50.

FIG. 3 is a simplified schematic diagram of acts of acomputer-implemented method 100 of processing a measured ultrasonicresponse waveform to determine a well casing thickness and an acousticimpedance of a sealing medium surrounding the well casing, in accordancewith many embodiments. The method 100 can optionally include processinga measured response waveform to determine an arrival time for a firstreflection from the well casing inner surface 32 (act 102). For example,referring to FIG. 2, the measured response waveform 50 can be processedto determine the point in time corresponding to the peak in thezero-order reflection transient 52. The arrival time can be used todetermine a distance between the transducer 26 and the well casing innersurface 32. In embodiments, the distance between the transducer 26 andthe inner surface 32, as well as displacement of the transducer 26 fromits nominal position at the center of the well casing 18, is used tocorrect for two-dimensional effects when generating candidate simulatedresponse waveforms.

The method 100 can optionally include processing the measured responsewaveform 50 to determine the approximate thickness of the well casing 18(act 104). For example, as illustrated in FIG. 4, the measured responsewaveform 50 can be processed to generate a frequency spectrum 106 thatcan be processed to identify a characteristic frequency 108. Thecharacteristic frequency 108 corresponds to the time of doublepropagation of the ultrasonic signal within the well casing 18. In manyembodiments, the characteristic frequency 108 is used to estimatethickness of the well casing 18.

In act 110, the measured response waveform 50 is processed to determinea first-iteration transducer impulse response. For example, referring toFIG. 5, an initial portion 112 of the measured response waveform 50 canbe selected as the first-iteration transducer impulse response. Anysuitable initial portion of the measured response waveform, for example,the first 34 to 38 time points in a 120 time point waveform, can beselected as the transducer impulse response. Alternatively, thetransducer impulse response can be measured experimentally using areflection from a thick and flat metal plate emulating an infinite halfspace.

In act 114, a first-iteration array of simulated response waveforms isgenerated based on the transducer impulse response, a first-iterationcandidate set of well casing thicknesses, a first-iteration candidateset of acoustic impedances for the surrounding sealing medium 24, afirst-iteration set of candidate annulus thicknesses based on theexpected thickness of the sealing layer, and a set of suitable candidatevalues for the acoustic impedance for the ground formation 16surrounding the annulus. For example, the candidate well casingthickness can be selected to cover a range that includes an expectednominal thickness for the particular well casing thickness or can bebased on an approximate casing thickness determined by processing themeasures response waveform as in act 104. Likewise, any suitable set ofvalues can be used for the first-iteration set of candidate 50 sealingmedium acoustic impedances. For example, the first-iteration set ofcandidate sealing medium acoustic impedances can be selected to cover asuitable range of acoustic impedances from an acoustic impedancecorresponding to a well-sealed annulus to an acoustic impedancecorresponding to the lack of sealing medium. Similarly, thefirst-iteration set of candidate annulus thicknesses can be based on theexpected thickness of the sealing layer and a suitable set of values canbe used for the acoustic impedance for the ground formation 16 based onthe known geological formation at any depth and azimuth

In many cases, the annulus can be considered thick enough that echoesfrom the annulus sealing medium 24/ground formation 16 interface 22 canbe ignored. For these cases, the parameters used in the matchingprocedure described below can be the casing thickness and the acousticimpedance of the sealing medium 24 in the annulus surrounding the casing18. In the description that follows, details of the matching procedureare presented using a two parameter model for ease in understandingsince the ultimate outputs of the processing described herein are robustestimates of the well casing thickness and the acoustic impedance forthe sealing medium 24 in the annulus around the well casing.Nevertheless, all steps apply equally to a four parameter model thatalso includes the thickness of the sealing medium 24 and the acousticimpedance of the ground formation 16 for those cases where the influenceof signals from the “third interface” 22 is significant.

Any suitable approach can be used to generate each of the simulatedresponse waveforms based on the respective candidate well casingthickness and the respective candidate surrounding sealing mediumacoustic impedance. For example, a simulated response waveform can begenerated by convolving the transducer impulse response with acalculated system response waveform. Any suitable calculated systemresponse waveform can be used. For example, the ultrasonic incidentpulse generated by the transducer 26 can be assumed to be adelta-function and the calculated system response waveform calculatedfor the delta function. FIG. 6 shows an example calculated systemresponse waveform 116. In many embodiments, the reflection transients118 in the calculated system response waveform 116 are spaced to reflectthe candidate well casing thickness and the amplitudes of the reflectiontransients in the calculated system response waveform 116 are calculatedto reflect the candidate surrounding sealing medium acoustic impedance.In embodiments, both the amplitude and pulse shape of the reflectiontransients 118 are adjusted to account for two-dimensional effectsrelated to the exact position of the transducer 26 with respect to theinner surface 32 of the casing, including diffraction, refraction, wellgeometry, and the resulting beam size at the transducer plane.Alterations in both the amplitude and pulse shape of reflectiontransients 118 in a one-dimensional signal model as illustrated in FIG.6 may be referred to herein as “1.3-D processing.” FIG. 7 shows anexample first-iteration simulated response waveform 118 generated byconvolving the transducer impulse response with the calculated systemresponse waveform 116.

In act 120, a first-iteration best-fit simulated response of thefirst-iteration array of simulated response waveforms is identified soas to identify a first-iteration best-fit well casing thickness and afirst-iteration best-fit sealing medium impedance. For example, each ofany suitable number of the array of first-iteration simulated responsewaveforms can be compared to the measured response waveform via aleast-squares comparison to select one of the first-iteration array ofsimulated response waveforms that best-fits the measured responsewaveform. FIG. 8 shows a plot that includes the example measuredresponse waveform 50 and the simulated response waveform 118. Thecandidate well casing thickness and the candidate sealing mediumacoustic impedance corresponding to the first-iteration best-fitsimulated response can be identified as being the well casing thicknessand the sealing medium acoustic impedance for the measured responsewaveform.

A second iteration can be performed using a corrected transducer impulseresponse to generate a second-iteration array of candidate simulatedresponse waveforms that may more closely match the measured responsewaveform due to reduced errors induced by the estimated transducerimpulse response used. FIG. 9 is a simplified schematic diagram of actsfor a second iteration that can be performed in the method 100, inaccordance with many embodiments. In act 122, a differential signal(Difference(t)) is calculated between the measured response waveform 50and the first-iteration best-fit simulated response waveform. In act124, a corrected impulse response can be determined as:IR ₁(t)=IR(t)−Difference(t)  equation (4)

where Difference(t) is the differential signal, i.e. the differencebetween the first-iteration best-fit simulated response waveform and themeasured response waveform 50. In act 126, a second-iteration array ofsimulated response waveforms is generated based on the second-iterationtransducer impulse response, a second-iteration set of candidate set ofwell casing thicknesses (which can be the same or different that thefirst-iteration set of candidate well casing thicknesses), and asecond-iteration set of acoustic impedances for the sealing medium(which can be the same or different from the first-iteration set ofcandidate sealing medium acoustic impedances). The second-iterationarray of simulated response waveforms can be generated using a similarapproach as described herein for the first-iteration array of simulatedresponse waveforms. In act 128, a second-iteration best-fit simulatedresponse of the second-iteration array of simulated response waveformsis identified so as to identify a second-iteration best-fit well casingthickness and a second-iteration best-fit sealing medium impedance. Thesecond-iteration best-fit simulated response can be identified using asimilar approach as described herein for identifying the first-iterationbest-fit simulated response.

FIG. 10A is a plot that includes the example measured response waveform50, the example first-iteration simulated response waveform 118, and asecond-iteration simulated response waveform 124 based on the correctedimpulse response. As can be seen, the second-iteration simulatedresponse waveform 124 matches the measured response waveform 50 betterthan the first-iteration simulated response waveform 118. Althoughsimilar additional iterations can be performed, almost no difference wasobtained for a third iteration compared with a second iteration. Oncethe second-iteration array of simulated response waveforms has beengenerated, the second-iteration best-fit simulated response can beidentified using a similar approach as for the first iteration.

FIG. 10B and FIG. 10C show overlays comparing simulated responsewaveforms for an infinitely thick cement-filled annulus surrounding awell casing and cement-filled annuluses having two different finitethicknesses, in accordance with many embodiments. When the thickness ofthe sealing medium is thin enough such that reflections from the sealingmedium 24/ground formation 16 interface 22 significantly affect themeasured waveform 50, reflections from the sealing medium 24/groundformation 16 interface 22 can be accounted for during generation ofsimulated waveforms matched to the measured response waveform 50.Reflections from the sealing medium 24/ground formation 16 interface 22may produce an additional chain of signals, which may be referred toherein as a “third interface signal”. The third interface signal may bedirectly related to reverberations of the probe signal in the casingbefore or after it is reflected from the third interface. The thirdinterface signal can be superposed with the primarily signal created bythe reverberation of the probe signal within the casing and produces analtered tail in the echo train depending on the thickness of theannulus. The resulting simulated signal tail (shown for two differentcement thicknesses in FIG. 10B and FIG. 10C) can be generated via asuperposition of the primarily signal with the third interface signal,where the characteristics of the tail depend on the sealing mediumthickness, sealing medium impedance, and the impedance of the groundformation surrounding the sealing medium. As seen in FIG. 10B for a16.88 mm thick annulus filled with cured cement, the tail (QQ) of theecho train is actually higher than that of an infinite cement layer. Ifinterpreted with a simple two-parameter model, the computed impedance ofthe sealing medium 24 will be significantly underestimated. As seen inFIG. 10C for a 29.19 mm thick annulus filled with cured cement, the tail(RR) is significantly altered compared to that for the infinite cementlayer. Because later echoes in the train for the 29.19 mm case attenuatemore slowly, a two-parameter model would again produce an underestimatedimpedance for the sealing medium 24 in the annulus. Overall, when theannulus is thin, the third interface reflection can be taken intoaccount to better estimate the acoustic impedance of the sealing medium24 in the annulus.

The approaches described herein can be repeated to process measuredresponse waveforms resulting from the inspection of any suitable numberof inspection points in a well. For example, in many embodiments, thedown-hole inspection tool 14 is configured to rotate the ultrasonictransducer 26 to generate measured response waveforms for differentazimuth angles and to be moved vertically within the well casing 18 togenerate measured response waveforms for different well depths. Each ofthe resulting array of measured response waveforms can be processed asdescribed herein to generate corresponding best-fit well casingthickness values and best-fit surrounding sealing medium acousticimpedance values. Any suitable approach can be used to output theresulting values. For example, FIG. 11 shows a resulting contour plot ofsurrounding sealing medium acoustic impedance for a range of depths andazimuth angles based on plane wave approximation (i.e., one-dimensional)based simulated response waveforms.

The simulated response waveforms can also be generated to account fortwo-dimensional effects so as to better account for ultrasonicpropagation relative to a plane-wave approximation. FIG. 12 illustrateshow two-dimensional effects associated with the cylindrical well casingcan change the direction of the waves such that higher-order reflectionsmiss the transducer 26. In particular, the pulse shape and amplitude ofeach reflection can be affected by diffraction and refraction effects.

FIG. 13 is a simplified schematic diagram of acts for generating asimulated response waveform that accounts for two-dimensional effects,in accordance with many embodiments. In act 130, amplitudes of simulatedecho waves are calculated. The amplitudes of the simulated echo wavescan be calculated using a plane-wave approximation. In act 132,corrected amplitudes and pulse shapes of the individual pulses areobtained by accounting for diffraction, refraction, and the resultingsize of the ultrasound beam at the transducer plane, as describedherein. Alterations in both the amplitude and pulse shape of reflectiontransients 118 in a one dimensional signal model as illustrated in FIG.6 may be referred to herein as “1.3-D processing.” Any suitable approachcan be used to correct the amplitudes and pulse shapes such as describedherein. In act 134, a first-iteration simulated response waveform isgenerated based on the corrected amplitudes and pulse shapes. In act136, the first-iteration simulated response waveform is convolved withthe transducer impulse response to generate a first-iteration simulatedresponse waveform that accounts for two-dimensional effects.

FIG. 14 is a simplified schematic diagram of acts of acomputer-implemented method 200 of generating a simulated responsewaveform that accounts for two-dimensional effects, in accordance withmany embodiments. The computer-implemented method 200 includes (a)reading in the measured response waveform (act 202), (b) setting programparameters (act 204), (c) performing spectral analysis of a measuredresponse waveform (act 206), (d) creating a first-iteration array ofsimulated response waveforms so as to account for two-dimensionaleffects (act 208), (e) identifying which of the first-iterationsimulated response waveforms best matches the measured response waveform(act 210), (f) performing a second iteration using a correctedtransducer impulse response (act 212), (g) calculating a confidencevalue (act 214), and (h) generating output (act 216).

In act 204, any suitable parameter can be set for use in processing themeasured response waveform using the approaches described herein. Forexample, the program parameters can include material densities,ultrasound speed in mud, ultrasound speed in the well casing 18,transducer diameter, distance from the transducer 26 to the well casing18, inspection depth range, inspection azimuth range, range of measuredresponse waveforms to process, determination of the transducer impulseresponse, which, as described herein, can initially be assumed toapproximate an initial portion of a measured response waveform that doesnot include waves generated by reverberations of the ultrasonic impulsewithin the well casing.

In act 206, spectral analysis of the measured response waveform can beused to determine the resonant frequency of the ultrasonicreverberations within the well casing. The resonant frequency can thenbe used to estimate thickness of the well casing. The signal arrivaltime can be determined by finding the maximum of the Hilbert transformed(i.e., detected) signal.

In act 208, the first-iteration array of candidate simulated responsewaveform can be created as described herein. Reflection and transmissioncoefficients can be defined at all the interfaces, including the ‘thirdinterface’ 22 between the sealing medium 24 and the ground formation 16.Initial echo amplitudes can be calculated. The initial echo amplitudesand pulse shapes in the echo train 118 shown in FIG. 6 can be correctedusing a numerical simulation of multi-dimensional wave propagation fromthe transducer 26 based on the position of the transducer 26 relative tothe front casing surface 32. A finite element simulation of wavepropagation can predict pulse shape changes of each pulse in the echotrain 118 compared to the assumed delta function excitation of thetransducer. Similarly, the amplitude of each echo in the echo train 118can be corrected according to the parabolic diffraction equation,refraction coefficients, and the resulting beam diameter at thetransducer 26. Alterations in both the amplitude and pulse shape ofreflection transients 118 in a one dimensional signal model asillustrated in FIG. 6 may be referred to herein as “1.3-D processing.”

Using the parabolic diffraction equation:

$\begin{matrix}{{{{\left( {\overset{->}{\tau}\;{\overset{->}{\nabla}{+ \frac{i}{2}}}\frac{d^{2}k}{d\;\omega^{2}}\frac{\partial^{2}}{\partial\tau^{2}}} \right)A} - {\frac{i}{2k}\Delta}}\bot A} = 0} & {{equation}\mspace{14mu}(5)}\end{matrix}$

where k is the wave number and A is the complex amplitude, the solutionfor Gaussian beams for the amplitude, beam radius, and wavefrontcurvature gives:

$\begin{matrix}{{A(z)} = {\frac{A(0)}{\sqrt{1 + D^{2}}}{\exp\left\lbrack {- \frac{r^{2}}{a_{0}^{2}\left( {1 + D^{2}} \right)}} \right\rbrack}}} & {{equation}\mspace{14mu}(6)} \\{{a^{2}(z)} = {a_{0}^{2}\left( {1 + D^{2}} \right)}} & {{equation}\mspace{14mu}(7)} \\{\frac{1}{R(z)} = {\frac{1}{a_{0}}\frac{\sqrt{D}}{\sqrt{1 + D^{2}}}}} & {{equation}\mspace{14mu}(8)}\end{matrix}$

where r=√{square root over (x²+y²)} is the distance from the transducernormal, a₀ is the transducer radius, and D=z/l_(d), l_(d)=πa₀ ²/C_(s)fis the diffraction length. The simulated response waveform can begenerated as a train of echo signals. Initially, each echo can have atemporal profile of a very short Gaussian signal (mimickingdelta-function):

$\begin{matrix}{{{{Gauss}_{n}\left( t_{n} \right)} = {A_{n}{\exp\left( {- \frac{\left( {t - t_{n}} \right)^{2}}{\tau_{0}^{2}}} \right)}}},} & {{equation}\mspace{14mu}(9)}\end{matrix}$

where τ₀ is equal to digitization time of the measured responsewaveform, t_(n)=2 nH/C_(s) is the arrival time of n_(th) echo, andamplitude A_(n) is equal to the reflection coefficient corresponding tothe respective echo adjusted by the correction of equation (6). TheGaussian temporal profile for each pulse can be adjusted based on theresults of computational models of wave propagation that includes theeffects of diffraction and refraction given the transducer positionrelative to the front casing surface 32 to produce the modified echopulse Echo_(n)(t_(n)). Thus:S _(sim)(t)=Σ_(n=1) ^(N) ^(ech) Echo_(n)(t _(n)),  equation (10)

where N_(ech) is a number of echoes in the measured response waveform.The resulting chain of echoes can be convolved then with the transducerimpulse response:R _(sim)(t)=S _(sim)(t)

IR(t)  equation (11)

to finalize the creation of the simulated response waveform for therespective combination of candidate well casing thickness, candidatesurrounding sealing medium acoustic impedance, and when appropriate,candidate annulus thickness and candidate acoustic impedance for thesurrounding ground formation. The process can be repeated for eachrespective combination of candidate well casing thickness, candidatesurrounding sealing medium acoustic impedance, candidate annulusthickness and candidate acoustic impedance for the surrounding groundformation to generate the first-iteration array of candidate simulatedresponse waveforms.

A simulated response waveform can be generated for each combination ofcandidate parameters (e.g., well casing thickness, surrounding sealingmedium acoustic impedance, annulus thickness and acoustic impedance forthe ground formation). In many embodiments, the surrounding sealingmedium is cement and the cement acoustic impedance is set in the rangeof 0 to 10 Mrayl with a step of 0.1 Mrayl. The casing thickness can bevaried about a nominal casing thickness (e.g., a nominal casingthickness of the well casing being inspected, an approximate casingthickness determined via the spectral analysis of the measured responsewaveform as described herein). The casing thickness can be varied aboutthe nominal casing thickness in any suitable step (e.g., about 1% of thenominal casing thickness). When appropriate, the annular thickness canbe varied about the nominal annular thickness in any suitable step(e.g., about 1% of the nominal casing thickness) and the acousticimpedance for the ground formation can be varied about the nominal valuein any suitable step (e.g., about 1% of an anticipated nominal acousticimpedance of the ground formation).

In act 210, the first-iteration casing thickness, first-iterationacoustic impedance of the sealing medium, first-iteration annulusthickness and first-iteration acoustic impedance for the groundformation can be determined by solving the inverse problem to find whichof the first-iteration array of simulated response waveforms bestmatches the measured response waveform. A matrix of integral time-gainnormalized mean-squared difference between the simulated responsewaveforms and the measured response waveform can be calculated for thewhole array of simulated parameters:Diff_(Nsim)=Σ_(i=1) ^(N) ^(points) (R _(sim)(i)−S_(ex)(i))²/Att²,  equation (12)

where Att(i) is the effective attenuation of the reflection echo train,to scale for the decaying waveform amplitude and make the contributionfrom all echoes to the total difference approximately equal. The bestmatched simulated response waveform is assumed to be based on the valuesof casing thickness and sealing material acoustic impedance that bestmatch actual values.

In act 212, the second iteration can be performed using the correctedtransducer impulse response. The differential signal S_(dif)(i)=R_(sim)(i)−S_(ex)(i) between the first-iteration best-fit simulatedresponse waveform and the measured response waveform can be calculatedfor the best matched parameter set. The temporal profile of thetransducer response used in the first-iteration can be corrected by thedifferential signal. Solution of the inverse problem is repeated for thesecond-iteration array of simulated response waveforms, which can begenerated using the corrected transducer impulse response.

In acts 214 and 216, the set of results can include (a) atwo-dimensional plot (e.g., a shaded contour plot, a contour line plot)of surrounding sealing medium acoustic impedances Zc obtained for alldepth and azimuthal points (e.g., as shown in FIG. 15); (b) atwo-dimensional plot (e.g., shaded contour plot, a contour line plot) ofcasing thicknesses H for all depth and azimuthal points; and (c) aconfidence matrix indicative of how closely each of best-fit simulatedresponse waveforms corresponds to the respective measured responsewaveform by, for example, calculating confidence values equal to theinverse of the integral of the time-gain normalized mean-squareddifference between the respective best-fit simulated response waveformand respective measured response waveform for each depth and azimuthalposition. Regions of low confidence indicate areas where the measuredsurrounding sealing medium acoustic impedance values may not bereliable. Regions of high confidence indicate areas where the measuredsurrounding sealing medium acoustic impedance values are likely to bereliable.

In addition to the confidence values, more complex measures ofconfidence can be obtained via a multivariable analysis of the measuredsurrounding sealing medium acoustic impedance, the measured well casingthickness, and the confidence value. For example, in addition to rawimpedance, thickness, and confidence value maps, spatial gradients ofthese parameters in both depth and azimuthal directions can be computed.By setting “soft” thresholds on these parameters (values themselves aswell as gradients) using sigmoid functions, the final confidence can becomputed using a fuzzy logic approach where simple logic processing onthe outputs of the soft thresholds can be used to determine a finalconfidence value.

In addition to displaying the confidence value independent of theacoustic impedance of the sealing medium, the two estimates can becombined into a simple image, where the confidence is used to modulatethe estimated sealing medium acoustic impedance value. For example, thesealing medium acoustic impedance can be displayed using a pure Chromascale and the confidence can be used to set the absolute brightness(luminance) at each pixel. In this way, low confidence regions aresimply black whereas high confidence regions show the sealing mediumacoustic impedance values in color. There are many variations on thistheme where confidence and sealing medium acoustic impedance values canbe combined into a single optimized display.

Some or all of the methods described herein (or any other processesdescribed herein, or variations, and/or combinations thereof) may beperformed under the control of one or more computer systems configuredwith executable instructions and may be implemented as code (e.g.,executable instructions, one or more computer programs, or one or moreapplications) executing collectively on one or more processors, byhardware or combinations thereof. The code may be stored on acomputer-readable storage medium, for example, in the form of a computerprogram comprising a plurality of instructions executable by one or moreprocessors. The computer-readable storage medium may be non-transitory.

Other variations are within the spirit of the present invention. Thus,while the invention is susceptible to various modifications andalternative constructions, certain illustrated embodiments thereof areshown in the drawings and have been described above in detail. It shouldbe understood, however, that there is no intention to limit theinvention to the specific form or forms disclosed, but on the contrary,the intention is to cover all modifications, alternative constructions,and equivalents falling within the spirit and scope of the invention, asdefined in the appended claims.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing the invention (especially in the context of thefollowing claims) are to be construed to cover both the singular and theplural, unless otherwise indicated herein or clearly contradicted bycontext. The terms “comprising,” “having,” “including,” and “containing”are to be construed as open-ended terms (i.e., meaning “including, butnot limited to,”) unless otherwise noted. The term “connected” is to beconstrued as partly or wholly contained within, attached to, or joinedtogether, even if there is something intervening. Recitation of rangesof values herein are merely intended to serve as a shorthand method ofreferring individually to each separate value falling within the range,unless otherwise indicated herein, and each separate value isincorporated into the specification as if it were individually recitedherein. All methods described herein can be performed in any suitableorder unless otherwise indicated herein or otherwise clearlycontradicted by context. The use of any and all examples, or exemplarylanguage (e.g., “such as”) provided herein, is intended merely to betterilluminate embodiments of the invention and does not pose a limitationon the scope of the invention unless otherwise claimed. No language inthe specification should be construed as indicating any non-claimedelement as essential to the practice of the invention.

Preferred embodiments of this invention are described herein, includingthe best mode known to the inventors for carrying out the invention.Variations of those preferred embodiments may become apparent to thoseof ordinary skill in the art upon reading the foregoing description. Theinventors expect skilled artisans to employ such variations asappropriate, and the inventors intend for the invention to be practicedotherwise than as specifically described herein. Accordingly, thisinvention includes all modifications and equivalents of the subjectmatter recited in the claims appended hereto as permitted by applicablelaw. Moreover, any combination of the above-described elements in allpossible variations thereof is encompassed by the invention unlessotherwise indicated herein or otherwise clearly contradicted by context.

What is claimed is:
 1. A computer-implemented method of processing ameasured ultrasonic response waveform to determine a thickness of a wellcasing thickness and an acoustic impedance of a sealing medium in anannulus surrounding the well casing, the method comprising: transmittingan ultrasonic signal from an ultrasonic inspection tool toward the wellcasing from within the well casing; generating, via an ultrasonictransducer, a measured response waveform resulting from the transmittedultrasonic signal, wherein the ultrasonic inspection tool comprises theultrasonic transducer; processing the measured response waveform todetermine an arrival time for a first reflection from the well casing;generating a plurality of simulated response waveforms corresponding tothe arrival time, a set of candidate acoustic impedances for the sealingmedium in the annulus, a set of candidate well casing thicknesses, a setof candidate annulus thicknesses, and a set of candidate acousticimpedances for a ground formation surrounding the annulus; identifying abest-fit simulated response waveform from the plurality of simulatedresponse waveforms that best matches the measured response waveform soas to identify a best fit well casing thickness of the set of candidatewell casing thicknesses, a best-fit sealing medium acoustic impedance ofthe set of candidate acoustic impedances for the medium in the annulus,a best-fit annulus thickness of the set of candidate annulusthicknesses, and a best-fit acoustic impedance for the ground formationsurrounding the annulus of the set of candidate acoustic impedances forthe ground formation surrounding the annulus associated with thebest-fit simulated response waveform; and outputting the best fitsealing medium acoustic impedance to a user via an output device.
 2. Thecomputer-implemented method of claim 1, further comprising: processing aportion of the measured response waveform to determine an approximatethickness of the well casing; and selecting the set of candidate wellcasing thicknesses based on the approximate thickness of the wellcasing.
 3. The computer-implemented method of claim 1, furthercomprising processing a portion of the measured response waveform todetermine a transducer impulse response for an ultrasonic transducerused to generate the measured response waveform, and wherein each of theplurality of simulated response waveforms is based on the transducerimpulse response.
 4. The computer-implemented method of claim 3, whereingenerating each of the plurality of simulated response waveformscomprises: calculating amplitudes for simulated echo waves returning tothe ultrasonic transducer from the well casing and an interface betweenthe sealing medium in the annulus and the ground formation surroundingthe annulus; generating corrected amplitudes and pulse shapes for thesimulated echo waves by correcting the amplitudes and pulse shapes forthe simulated echo waves to account for multi-dimensional propagationeffects including diffraction, refraction, and resulting beam size at atransducer plane of the ultrasonic transducer; generating a firstsimulated waveform based on the corrected amplitudes and pulse shapes;and convolving the first simulated waveform with the transducer impulseresponse to generate the respective simulated response waveform.
 5. Thecomputer-implemented method of claim 3, further comprising: calculatinga differential signal between the measured response waveform and thebest-fit simulated response waveform; calculating a corrected transducerimpulse response for the ultrasonic transducer by correcting thetransducer impulse response based on the differential signal; generatinga second-iteration plurality of simulated response waveforms based onthe corrected transducer impulse response and corresponding to thearrival time, a second-iteration set of candidate acoustic impedancesfor the sealing medium surrounding the well casing, a second-iterationset of candidate well casing thicknesses, a second-iteration set ofcandidate annulus thicknesses, and a second-iteration set of candidateacoustic impedances for the ground formation surrounding the annulus;and identifying a second-iteration best-fit simulated response waveformfrom the second-iteration plurality of simulated response waveforms thatbest matches the measured response waveform so as to identify asecond-iteration best-fit well casing thickness of the second-iterationset of candidate well casing thicknesses, a second-iteration best-fitsealing medium acoustic impedance of the second-iteration set ofcandidate acoustic impedances for the sealing medium surrounding thewell casing, a second-iteration best-fit annulus thickness of thesecond-iteration set of candidate annulus thicknesses, and asecond-iteration best-fit acoustic impedance for the ground formationsurrounding the annulus of the second-iteration set of candidateacoustic impedances for the ground formation surrounding the annulusassociated with the second-iteration best-fit simulated responsewaveform.
 6. The computer-implemented method of claim 5, whereingenerating each of the second-iteration plurality of simulated responsewaveforms comprises convolving a simulated response waveform with thecorrected transducer impulse response to generate the respectivesimulated response waveform of the second-iteration plurality ofsimulated response waveforms.
 7. The computer-implemented method ofclaim 1, wherein each of the plurality of simulated response waveformsis based on reflection of the ultrasonic signal from an inner surface ofthe well casing, reflection of the ultrasonic signal from an interfacebetween the well casing and the sealing medium in the annulus,reflection of the ultrasonic signal from an interface between thesealing medium in the annulus and the ground formation surrounding theannulus, and multiple reverberations of the ultrasonic signal within thewell casing.
 8. The computer-implemented method of claim 1, furthercomprising generating a confidence value based on deviation between thebest-fit simulated response waveform and the measured response waveform.9. The computer-implemented method of claim 8, further comprisinganalyzing the best-fit well casing thickness, the best-fit sealingmedium acoustic impedance, and the confidence value in combination toproduce a final confidence value indicative of whether the actualsealing medium acoustic impedance is within a designated range.
 10. Thecomputer-implemented method of claim 8, further comprising calculatingthe best-fit well casing thickness, the best-fit sealing medium acousticimpedance, and the final confidence value for a plurality of depths anda plurality of azimuth angles.
 11. A system for processing a measuredultrasonic response waveform to determine a thickness of a well casingand an acoustic impedance of a sealing medium in an annulus surroundingthe well casing, the system comprising: an ultrasonic inspection toolthat transmits an ultrasonic signal toward the well casing from withinthe well casing, the ultrasonic inspection tool comprising an ultrasonictransducer that generates a measured response waveform resulting fromthe transmitted ultrasonic signal; an output device; a processor; atangible memory storing non-transient instructions executable by theprocessor to cause the processor to: process a measured responsewaveform resulting from an ultrasonic signal transmitted toward the wellcasing from within the well casing to determine an arrival time for afirst reflection from the well casing; generate a plurality of simulatedresponse waveforms corresponding to the arrival time, a set of candidateacoustic impedances for the sealing medium in the annulus, a set ofcandidate well casing thicknesses, a set of candidate annulusthicknesses, and a set of candidate acoustic impedances for a groundformation surrounding the annulus; identify a best-fit simulatedresponse waveform from the plurality of simulated response waveformsthat best matches the measured response waveform so as to identify abest-fit well casing thickness of the set of candidate well casingthicknesses, a best-fit sealing medium acoustic impedance of the set ofcandidate acoustic impedances for the sealing medium in the annulus, abest-fit annulus thickness of the set of candidate annulus thicknesses,and a best-fit acoustic impedance for the ground formation surroundingthe annulus of the set of candidate acoustic impedances for the groundformation surrounding the annulus associated with the best-fit simulatedresponse waveform; and output, via the output device, the best-fitsealing medium acoustic impedance for display to a user of the system.12. The system of claim 11, wherein the instructions are furtherexecutable by the processor to: process a portion of the measuredresponse waveform to determine an approximate thickness of the wellcasing; and select the set of candidate well casing thicknesses based onthe approximate thickness of the well casing.
 13. The system of claim11, wherein the instructions are further executable by the processor tocause the processor to process a portion of the measured responsewaveform to determine a transducer impulse response for an ultrasonictransducer used to generate the measured response waveform, and whereineach of the plurality of simulated response waveforms is based on thetransducer impulse response.
 14. The system of claim 13, wherein theinstructions are executable by the processor to cause the processor to,for each of the plurality of simulated response waveforms correspondingto the arrival time, calculate amplitudes for simulated echo wavesreturning to the ultrasonic transducer from the well casing; generatecorrected amplitudes and pulse shapes for the simulated echo waves bycorrecting the amplitudes and pulse shapes for the simulated echo wavesto account for multi-dimensional propagation effects includingdiffraction, refraction, well geometry, and a resulting beam size at atransducer plane of the ultrasonic transducer; generate a firstsimulated waveform based on the corrected amplitudes and pulse shapes;and convolve the first simulated waveform with the transducer impulseresponse to generate the respective simulated response waveform.
 15. Thesystem of claim 13, wherein the instructions are executable by theprocessor to: calculate a differential signal between the measuredresponse waveform and the best-fit simulated response waveform;calculate a corrected transducer impulse response for the ultrasonictransducer by correcting the transducer impulse response based on thedifferential signal; generate a second-iteration plurality of simulatedresponse waveforms based on the corrected transducer impulse responseand corresponding to the arrival time, a second-iteration set ofcandidate acoustic impedances for the sealing medium surrounding thewell casing, a second-iteration set of candidate well casing thicknesseswithin a range that includes the approximate thickness of the wellcasing, a second-iteration set of candidate annulus thicknesses within arange that includes an approximate thickness of the annulus, and asecond-iteration set of candidate acoustic impedances for the groundformation surrounding the annulus; and identify a second-iterationbest-fit simulated response waveform from the second-iteration pluralityof simulated response waveforms that best matches the measured responsewaveform so as to identify a second-iteration best-fit well casingthickness of the second-iteration set of candidate well casingthicknesses, a second-iteration best-fit sealing medium acousticimpedance of the second-iteration set of candidate acoustic impedancesfor the sealing medium in the annulus, a second-iteration best-fitannulus thickness of the second-iteration set of candidate annulusthicknesses, and a second-iteration best-fit acoustic impedance for theground formation surrounding the annulus of the second-iteration set ofcandidate acoustic impedances for the ground formation surrounding theannulus associated with the second-iteration best-fit simulated responsewaveform.
 16. The system of claim 15, wherein the instructions areexecutable by the processor to convolve a simulated response waveformwith the corrected transducer impulse response to generate therespective simulated response waveform of the second-iteration pluralityof simulated response waveforms.
 17. The system of claim 11, whereineach of the plurality of simulated response waveforms is based onreflection of the ultrasonic signal from an inner surface of the wellcasing, reflection of the ultrasonic signal from an interface betweenthe well casing and the sealing medium in the annulus, reflection of theultrasonic signal from an interface between the sealing medium in theannulus and the ground formation surrounding the annulus, and multiplereverberations of the ultrasonic signal within the well casing.
 18. Thesystem of claim 11, wherein the instructions are executable by theprocessor to generate a confidence value based on deviation between thebest-fit simulated response waveform and the measured response waveform.19. The system of claim 18, wherein the instructions are executable bythe processor to analyze the best-fit well casing thickness, thebest-fit sealing medium acoustic impedance, and the confidence value incombination to produce a final confidence value indicative of whetherthe actual sealing medium acoustic impedance is within a designatedrange.
 20. A computer-implemented method of processing a measuredultrasonic response waveform to determine a thickness of a well casingthickness and an acoustic impedance of a sealing medium in an annulussurrounding the well casing, the method comprising: transmitting anultrasonic signal from an ultrasonic inspection tool toward the wellcasing from within the well casing; generating, via an ultrasonictransducer, a measured response waveform resulting from the transmittedultrasonic signal, wherein the ultrasonic inspection tool comprises theultrasonic transducer; processing the measured response waveform todetermine an arrival time for a first reflection from the well casing;generating a plurality of simulated response waveforms corresponding tothe arrival time, a set of candidate acoustic impedances for the sealingmedium in the annulus and a set of candidate well casing thicknesses;identifying a best-fit simulated response waveform from the plurality ofsimulated response waveforms that best matches the measured responsewaveform so as to identify a best fit well casing thickness of the setof candidate well casing thicknesses and a best-fit sealing mediumacoustic impedance of the set of candidate acoustic impedances for thesealing medium in the annulus associated with the best-fit simulatedresponse waveform; and outputting the best fit sealing medium acousticimpedance to a user via an output device.